Queueing Networks and Markov Chains Modeling and Performance Evaluation with Computer Science Applications Second Edition... Queueing Networks and Markov Chains Modeling and Performanc
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and Markov Chains
Modeling and Performance Evaluation with Computer Science Applications Second Edition
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and Markov Chains
Modeling and Performance Evaluation with Computer Science Applications Second Edition
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Library of Congress Cataloging-in-Publication Data:
Queueing networks and Markov chains : modeling and performance evaluation with computer science applications / Gunter Bolch , [et al.].-2nd rcv and enlarged ed
“A Wiley-lnterscience publication.”
Includes bibliographical references and index
ISBN- I3 978-0-47 1-56525-3 (acid-free paper)
ISBN- I0 0-47 1-56525-3 (acid-free paper)
I Markov processes 2 Queuing theory I Bolch, Gunter
Trang 7Contents
Preface to the Second Edition X l l l Preface to the First Edition xv 1 Introduction 1 1.1 Motivation 1
1.2 Methodological Background 5
1.2.1 Problem Formulation 6
1.2.2 The Modeling Process 8
1.2.3 Evaluation 10
1.2.4 Summary 12
1.3 Basics of Probability and Statistics 15
1.3.1 Random Variables 15
1.3.2 Multiple Random Variables 30
1.3.3 Transforms 36
1.3.4 Parameter Estimation 38
1.3.5 Order Statistics 46
1.3.6 Distribution of Sums 46
V
Trang 8vi CONTENTS
2.1 Markov Processes 51
2.1.1 Stochastic and Markov Processes 51
2.1.2 Markov Chains 53
2.2.1 A Simple Example 71
2.2.2 Markov Reward Models 75
2.2.3 A Casestudy 80
2.3 Generation Methods 90
2.3.1 Petri Nets 94
2.3.2 Generalized Stochastic Petri Nets 96
2.3.3 Stochastic Reward Nets 97
2.3.4 GSPN/SRN Analysis 101
A Larger Exanlple 108
2.3.6 Stochastic Petri Net Extensions 113
2.3.7 Non-Markoviarl Models 115
2.3.8 Symbolic State Space Storage Techniques 120
2.2 Performance Measures 71
2.3.5 3 Steady-State Solutions of Markov Chains 123 3.1 Solution for a Birth Death Process 125
3.2 Matrix-Geometric Method: Quasi-Birth-Death Process 127
3.2.1 The Concept 127
3.2.2 Example: The QBD Process 128
3.3 Hessenberg Matrix: Non-Markovian Queues 140
3.3.1 Nonexporlential Servicc Times 141
3.3.2 Server with Vacations 146
3.4 Numerical Solution: Direct Methods 151
3.4.1 Gaussian Elimination 152
3.4.2 The Grassmanrl Algorithm 158
3.5 Numerical Solution: Iterative Methods 165
3.5.1 Convergence of Iterative Methods 165
3.5.2 Power Method 166
3.5.3 Jacobi's Method 169
3.5.4 Gauss-Seidel Method 172
3.5.5 The Method of Successive Over-Relaxation 173
3.6 Comparison of Numerical Solution Methods 177
3.6.1 Case Studies 179
Trang 94 Steady-S tate Aggregation/Disaggregation
4.1 Courtois’ Approximate Method 185
4.1.1 Decomposition 186
4.1.2 Applicability 192
4.1.3 Analysis of the Substructures 194
4.1.4 Aggregation and Unconditioning 195
4.1.5 The Algorithm 197
4.2 Takahashi’s Iterative Method 198
4.2.1 The Fundamental Equations 199
4.2.2 Applicability 201
4.2.3 The Algorithm 202
4.2.4 Application 202
4.2.5 Final Remarks 206
5 Transient Solution of Markov Chains 209 5.1 Transient Analysis Using Exact Methods 210
5.1.1 A Pure Birth Process 210
5.1.2 A Two-State CTMC 213
5.1.3 Solution Using Laplace Transforms 216
5.1.4 Numerical Solution Using Uniformization 216
5.1.5 Other Numerical Methods 221
5.2 Aggregation of Stiff Markov Chains 222
5.2.1 Outline arid Basic Definitions 223
5.2.2 Aggregation of Fast Recurretit Subset s 224
5.2.3 Aggregation of Fast Transient Subsets 227
5.2.4 Aggregation of Initial State Probabilities 228
5.2.5 Disaggregations 229
5.2.6 The Algorithm 230
5.2.7 An Example: Server Breakdown arid Repair 232
6 Single Station Queueing Systems 241 6.1 Notation 242
6.1.1 Kendall’s Notation 242
6.1.2 Performance Measures 244
6.2 Markovian Queues 246
6.2.1 The M/M/l Queue 246
Trang 10viii CONTENTS
6.2.2 The M/M/ca Queue 249
6.2.3 The M/M/m Queue 250
6.2.4 The M / M / l / K Finite Capacity Queue 251
6.2.5 Machine Repairman Model 252
6.2.6 Closed Tandem Network 253
6.3 Non-Markovian Queues 255
6.3.1 The M / G / 1 Queue 255
6.3.2 The GI/M/l Queue 261
6.3.3 The GI/M/m Queue 265
6.3.4 The GI/G/1 Queue 265
6.3.5 The M/G/m Queue 267
The GI/G/m Queue 269
6.4 Priority Queues 272
6.4.1 Queue without Preemption 272
6.4.2 Conservation Laws 278
6.4.3 Queur: with Preemption 279
6.4.4 Queue with Time-Dependent Priorities 280
6.5 Asymmetric Queues 283
6.5.1 Approximate Analysis 284
6.5.2 Exact Analysis 286
6.6 Queues with Batch Service and Batch Arrivals 295
6.6.1 Batch Service 295
6.6.2 Batch Arrivals 296
6.7 Retrial Queues 299
6.7.1 M/M/1 Ret rial Queue 300
6.7.2 M / G / l Retrial Queue 301
Special Classes of' Point Arrival Processes 302
6.8.1 Point, Renewal, and Markov Renewal Processes 303
6.8.2 MMPP 303
6.8.3 MAP 306
6.8.4 BMAP 309
6.3.6 6.8 7 Queueing Networks 321 7.1 Definit ions and Notation 323
7.1.1 Single Class Networks 323
7.1.2 Multiclass Networks 325
7.2 Performance Measures 326
7.2.1 Single Class Networks 326
Trang 11CONTENTS ix
7.2.2 Multiclass Networks 330
7.3 Product-Form Queueing Networks 331
7.3.1 Global Balance 332
7.3.2 Local Balance 335
7.3.3 Product-Form 340
7.3.4 Jackson Networks 341
7.3.5 Gordon-Newel1 Networks 346
7.3.6 BCMP Networks 353
8 Algorithms for Product-Form Networks 369 8.1 The Convolution Algorithm 371
8.1.1 Single Class Closed Networks 371
8.1.2 Multiclass Closed Networks 378
8.2 The Mean Value Analysis 384
8.2.1 Single Class Closed Networks 385
8.2.2 Multiclass Closed Networks 393
8.2.3 Mixed Networks 400
8.2.4 Networks with Load-Dependent Service 405
8.3 Flow Equivalent Server Method 410
8.3.1 FES Method for a Single Node 410
8.3.2 FES Method for Multiple Nodes 414
8.4 Summary 417
9 Approximation Algorithms for Product-Form Networks 421 9.1 Approximations Based on the MVA 422
9.1.1 Bard Schweitzer Approximation 422
9.1.2 Self-correcting Approximation Technique 427
9.2 Summation Method 440
9.2.1 Single Class Networks 442
9.2.2 Multiclass Networks 445
9.3 Bottapprox Method 447
9.3.1 Initial Value of X 447
9.3.2 Single Class Networks 447
9.3.3 Multiclass Networks 450
9.4 Bounds Analysis 452
9.4.1 Asymptotic Bounds Analysis 453
Trang 12x CONTENTS
9.4.2 Balanced Job Bounds Analysis 456
9.5 Summary 459
10 Algorithms for Non-Product-Form Networks 461 10.1 Nonexponential Distributions 463
10.1.1 Diffusion Approximation 463
10.1.2 Maximum Entropy Method 470
10.1.3 Decomposition for Open Networks 479
10.1.4 Methods for Closed Networks 488
10.1.5 Closing Method for Open and Mixed Networks 507
10.2 Different Service Times at FCFS Nodes 512
10.3 Priority Networks 514
10.3.1 PRIOMVA 514
10.3.2 The Method of Shadow Server 522
10.3.3 PRIOSUM 537
10.4 Simultaneous Resource Possession 541
10.4.1 Memory Constraints 541
10.4.2 1/0 Subsystems 544
10.4.3 Method of Surrogate Delays 547
10.4.4 Serialization 548
10.5 Programs with Internal Concurrency 549
10.6 Parallel Processing 550
10.6.1 Asynchronous Tasks 551
10.6.2 Fork-Join Systems 558
10.7 Networks with Asymmetric Nodes 577
10.7.1 Closed Networks 577
10.7.2 Open Networks 581
10.8 Networks with Blocking 591
10.8.1 Different Blocking Types 592
10.9 Networks with Batch Service 600
10.9.1 Open Networks with Batch Service 600
10.9.2 Closed Networks with Batch Service 602
10.8.2 Product-Form Solution for Networks with Two Nodes 593 11 Discrete-Event Simulation 607 11.1 Introduction to Simulat ion 607
11.2 Simulative or Analytic Solution? 608
Trang 13CONTENTS xi
11.3 Classification of Simulatiori Models 610
11.4 Classification of Tools in DES 612
11.5 The Role of Probability and Statistics in Simulation 613
11.5.1 Random Variate Generation 614
11.5.2 Generating Events from an Arrival Process 624
11.5.3 Output Analysis 629
11.5.4 Speedup Techniques 636
11.5.5 Summary of Output Analysis 639
11.6 Applications 639
11.6.1 CSIM-19 640
11.6.2 Web Cache Example in CSIM-19 641
11.6.3 OPNET Modeler 647
11.6.4 ns-2 651
11.6.5 Model Construction in ns-2 652
12 Performance Analysis Tools 657 12.1 PEPSY 658
12.1.1 Structure of PEPSY 659
12.1.2 Different Programs in PEPSY 660
12.1.3 Example of Using PEPSY 661
12.1.4 Graphical User Interface XPEPSY 663
12.1.5 WinPEPSY 665
12.2 SPNP 666
12.2.1 SPNP Features 668
12.2.2 The CSPL Language 669
12.2.3 iSPN 673
12.3 MOSEL-2 676
12.3.1 Introduction 676
12.3.2 The MOSEL-2 Formal Description Technique 679
12.3.3 Tandem Network with Blocking after Service 683
12.3.4 A Retrial Queue 685
12.3.5 Conclusions 687
12.4 SHARPE 688
12.4.1 Central-Server Queueing Network 689
12.4.2 M/M/m/K System 691
12.4.3 M/M/I/K System with Server Failure and Repair 693
12.4.4 GSPN Model of a Polling System 695
12.5 Characteristics of Some Tools 701
Trang 14xii CONTENTS
13.1 Case Studies of Queueing Networks 703
13.1.1 Multiprocessor Systems 704
13.1.2 Client-Server Systems 707
13.1.3 Communication Systems 709
13.1.4 Proportional Differentiated Services 720
13.1.5 UNIX Kernel 724
13.1.6 J2EE Applications 733
13.1.7 Flexible Production Systems 745
13.1.8 Kanban Control 753
13.2 Case Studies of Markov Chains 756
13.2.1 Wafer Production System 756
13.2.2 Polling Systems 759
13.2.3 Client-Server Systems 762
13.2.4 ISDN Channel 767
13.2.5 ATM Network IJnder Overload 775
13.2.6 UMTS Cell with Virtual Zones 782
13.2.7 Handoff Schemes in Cellular Mobile Networks 786
13.3 Case Studies of Hierarchical R4odels 793
13.3.1 A Multiprocessor with Different Cache Strategies 793
13.3.2 Performability of a Multiprocessor System 803
Trang 15Preface to the Second
Edition
Nearly eight years have passed since the publication of the first edition of this book In this second edition, we have thoroughly revised all the chap- ters Many examples and problems are updated, and many new examples and problems have been added A significant addition is a new chapter on simula- tion methods arid applications Application to current topics such as wireless system performance, Internet performance, J2EE applications, and Kanban systems performance are added New material on non-Markovian and fluid stochastic Petri nets, along with solution techniques for Markov regenerative processes, is added Topics that are covered briefly include self-similarity, large deviation theory, and diffusion approximation The topic of hierarchical and fixed-point iterative models is also covered briefly Our collective research experience and the application of these methods in practice for the past 30 years (at the time of writing) have been distilled in these chapters as much
as possible We hope that the book will be of use as a classroom textbook
as well as of use for practicing engineers Researchers will also find valuable information here
We wish to thank many of our current students and former postdoctoral associates:
0 Dr Jorg Barner, for his contribution of the methodological background section in Chapter 1 He again supported us a lot in laying out the chap- ters and producing and improving figures and plots and with intensive proofreading
Trang 16
xiv PREFACE TO THE SECOND EDITION
0 Pawan Choudhary, who helped considerably with the simulation chap- ter, Dr Dharmaraja Selvamuthu helped with the section on SHARPE, and Dr Hairong Sun helped in reading several chapters
0 Felix Engelhard, who was responsible for the new or extended sections
on distributions, parameter estimation, Petri nets, and non-Markovian systems He also did a thorough proofreading
0 Patrick Wuchner, for preparing the sections on matrix-analytic and ma- trix-geometric methods as well as the MMPP and MAP sections, and also for intensive proofreading
0 Dr Michael Frank, who wrote and extended several sections: batch system and networks, summation method, and Kanban systems
0 Lassaad Essafi, who wrote the application section on differentiated ser- vices in the Internet
Thanks are also due to Dr Samuel Kounev and Prof Alejandro Buchmann for allowing us to use their paper "Performance Modelling and Evaluation
of Large Scale J2EE Applications" to produce the J2EE section, which is a shortened and adapted version of their paper
Our special thanks are due to Prof Helena Szczerbicka for her invaluable contribution to Chapter 11 on simulation and to modeling methodology sec- tion of Chapter 1 Her overall help with the second edition is also appreciated
We also thank Val Moliere, George Telecki, Emily Simmons, and Whitney
A Lesch from John Wiley & Sons for their patience and encouragement The support from the Euro-NGI (Design and Engineering of the Next Gen- eration Internet) Network of Excellence, European Commission grant IST- 507613: is acknowledged
Finally, a Web page has been set up for further information regarding the second edition The URL is h t t p : //www net fmi uni-passau de/QNMC2/
Gunter Bolch, Stefan Greiner, Hermarin de Meer, Kishor S Trivedi
Trang 17Preface to the First
Edition
Queueing networks and Markov chains are commonly used for the perfor- mance and reliability evaluation of computer, communication, and manu- facturing systems Although there are quite a few books on the individual topics of queueing networks and Markov chains, we have found none that covers both of these topics The purpose of this book, therefore, is to offer a detailed treatment of queueing systems, queueing networks, and continuous and discrete-time Markov chains
In addition to introducing the basics of these subjects, we have endeav- ored to:
0 Provide some in-depth numerical solution algorithms
0 Incorporate a rich set of examples that demonstrate the application of the different paradigms and corresponding algorithms
0 Discuss stochastic Petri nets as a high-level description language, there-
by facilitating automatic generation and solution of voluminous Markov chains
0 Treat in some detail approximation methods that will handle large mod- els
0 Describe and apply four software packages throughout the text
0 Provide problems as exercises
This book easily lends itself to a course on performance evaluation in the computer science and computer engineering curricula It can also be used for a course 011 stochastic models in mathematics, operations research and industri-
al engineering departments Because it incorporates a rich and comprehensive set of numerical solution methods comparatively presented, the text may also
xv
Trang 18xvi PREFACE TO THE FlRST EDlTlON
well serve practitioners in various fields of applications as a reference book for algorithms
With sincere appreciation to our friends, colleagues, and students who so
ably and patiently supported our manuscript project, we wish to publicly acknowledge:
0 Jorg Barner and Stepban Kosters, for their painstaking work in keying the text and in laying out the figures and plots
0 Peter Bazan, who assisted both with the programming of many examples and comprehensive proofreading
0 Hana SevEikovg, who lent a hand in solving many of the examples and contributed with proofreading
0 Jdnos Sztrik, for his comprehensive proofreading
0 Doris Ehrenreich, who wrote the first version of the section on commu- nication systems
0 Markus Decker, who prepared the first draft of the mixed queueing networks sect ion
0 Those who read parts of the manuscript and provided many useful com- ments, including: Khalid Begain, Oliver Dusterhoft, Ricardo Fricks, Swapna Gokhale, Thomas Hahn Christophe Hirel, Graham Horton, Steve Hunter, Demetres Kouvatsos, Yue Ma, Raymond Marie, Var- sha Mainkar, Victor Nicola, Cheul Woo Ro, Helena Szczerbicka, Lorrie Tomek, Bernd Wolfinger, Katinka Wolter, Martin Zaddach, and Henry Zang
Gunter Bolch and Stefan Greiner are grateful t o Fridolin Hofmann, and Hermann de Meer is grateful to Bernd Wolfinger, for their support in providing the necessary freedom from distracting obligations
Thanks are also due to Teubner B.G Publishing House for allowing us to borrow sections from the book entitled Leistungsbewertung von Rechen- systemen (originally in German) by one of the coauthors, Gunter Bolch In the present book, these sections are integrated in Chapters 1 and 7 through 10
We also thank Andrew Smith, Lisa Van Horn, and Mary Lynn of John Wiley & Sons for their patience and encouragement
The financial support from the SFB (Collaborative Research Centre) 182 (“Multiprocessor and Network Configurations”) of the DFG (Deutsche For- schungsgemeinschaft) is acknowledged
Finally, a Web page has been set up for further information regarding the book The URL is h t t p : //www4 cs f au de/QNMC/
GUNTER BOLCH, STEFAN GREINER, HERMANN DE MEER, KISHOR s TRIVEDI
Erlangen, June 1998
Trang 19Introduction
Information processing system designers need methods for the quantification
of system design factors such as performance and reliability Modern com- puter, communication, and production line systems process complex work- loads with random service demands Probabilistic and statistical methods are commonly employed for the purpose of performance and reliability eval- uation The purpose of-this book is to explore major probabilistic modeling techniques for the performance analysis of information processing systems Statistical methods are also of great importance but we refer the reader to other sources [Jaingl, TrivOl] for this topic Although we concentrate on per- formance analysis, we occasionally consider reliability, availability, and com- bined performance and reliability analysis Performance measures that are commonly of interest include throughput, resource utilization, loss probabili-
ty, and delay (or response time)
The most direct method for performance evaluation is based on actual measurement of the system under study However, during the design phase, the system is not available for such experiments, and yet performance of a given design needs to be predicted to verify that it meets design requirements and to carry out necessary trade-offs Hence, abstract models are necessary for performance prediction of designs The most popular models are based on discrete-event simulation (DES) DES can be applied to almost all problems
of interest, and system details to the desired degree can be captured in such
1
Trang 20simulation models Furthermore, many software packages are available that facilitate the construction and execution of DES models
The principal drawback of DES models, however, is the time taken to run such models for large, realistic systems, particularly when results with high accuracy (i.e., narrow confidence intervals) are desired A cost-effective alter- native to DES models, analytic models can provide relatively quick answers to
“what if” questions and can provide more insight into the system being stud- ied However, analytic models are often plagued by unrealistic assumptions that need to be made in order to make them tractable Recent advances in stochastic models and numerical solution techniques, availability of software packages, and easy access to workstations with large computational capa- bilities have extended the capabilities of analytic models to more complex systems
Analytical models can be broadly classified into state-space models and non-state-space models Most commonly used state-space models are Markov chains First introduced by A A Markov in 1907, Markov chains have been in use in performance analysis since around 1950 In the past decade, consider- able advances have been made in the numerical solution techniques, methods
of automated state-space generation, and the availability of software packages These advances have resulted in extensive use of Markov chains in performance and reliability analysis A Markov chain consists of a set of states and a set of labeled transitions between the states A state of the Markov chain can model various conditions of interest in the system being studied These could be the number of jobs of various types waiting to use each resource the number of resources of each type that have failed, the number of concurrent tasks of a given job being executed, and so on After a sojourn in a state, the Markov chain will make a transition to another state Such transitions are labeled with either probabilities of transition (in case of discrete-time Markov chains)
or rates of transition (in case of continuous-time Markov chains)
Long run (steady-state) dynamics of Markov chains can be studied using
a system of linear equations with one equation for each state Transient (or time dependent) behavior of a continuous-time Markov chain gives rise to
a system of first-order, linear, ordinary differential equations Solution of these equations results in state probabilities of the Markov chain from which desired performance measures can be easily obtained The number of states
in a Markov chain of a complex system can become very large, and, hence, automated generation and efficient numerical solution methods for underlying equations are desired A number of concise notations (based on queueing networks and stochastic Petri nets) have evolved, and software packages that automatically generate the underlying state space of the Markov chain are now available These packages also carry out efficient solution of steady-state and transient behavior of Markov chains In spite of these advances, there
is a continuing need to be able to deal with larger Markov chains and much research is being devoted to this topic
Trang 21MOTIVATION 3
If the Markov chain has nice structure, it is often possible to avoid the generation and solution of the underlying (large) state space For a class
of queueing networks, known as product-form queueing networks (PFQN),
it is possible to derive steady-state performance measures without resorting
to the underlying state space Such models are therefore called non-state- space models Other examples of non-state-space models are directed acyclic task precedence graphs [SaTr87] and fault-trees [STP96] Other examples of methods exploiting Markov chains with “nice” structure are matrix-geometric methods [Neut81] (see Section 3.2)
Relatively large PFQN can be solved by means of a small number of sinipler equations However, practical queueing networks can often get so large that approximate methods are needed to solve such PFQN Furthermore, many practical queueing networks (so-called non-product-form queueing networks, NPFQN) do not satisfy restrictions implied by product form In such cases,
it is often possible to obtain accurate approximations using variations of algo- rithms used for PFQNs Other approximation techniques using hierarchical and fixed-point iterative methods are also used
The flowchart shown in Fig 1.1 gives the organization of this book After a brief treatment on methodological background (Section 1.2), Section 1.3 covers the basics of probability and statistics In Chapter 2, Markov chains basics are presented together with generation methods for them Exact steady- state solution techniques for Markov chains are given in Chapter 3 and their
aggregation/disaggregation counterpart in Chapter 4 These aggregation/dis-
aggregation solution techniques are useful for practical Markov chain models with very large state spaces Transient solution techniques for Markov chains are introduced in Chapter 5
Chapter 6 deals with the description and coniputation of performance niea- sures for single-station queueing systems in steady state A general description
of queueing networks is given in Chapter 7 Exact solution methods for PFQN are described in detail in Chapter 8 while approximate solution techniques for PFQN are described in Chapter 9 Solution algorithms for different types
of NPFQN (such as networks with priorities, nonexponential service times, blocking, or parallel processing) are presented in Chapter 10
Since there are many practical problems that may not be analytically tractable, discrete-event simulation is commonly used in this situations We introduce the basics of DES in Chapter 11 For the practical use of modeling techniques described in this book, software packages (tools) are needed Chap- ter 12 is devoted to the introduction of a queueing network tool, a stochastic Petri net tool, a tool based on Markov chains and a toolkit with many mod-
el types, and the facility for hierarchical modeling is also introduced Each tool is described in some detail together with a simple example Throughout the book we have provided many example applications of different algorithms introduced in the book Finally, Chapter 13 is devoted to several large real-life applications of the modeling techniques presented in the book
Trang 23METHODOLOGICAL BACKGROUND 5
The focus of this book is the application of stochastic and probabilistic meth- ods to obtain conclusions about performance and reliability properties of a wide range of systems In general, a system can be regarded as a collection
of components which are organized and interact in order to fulfill a common task [IEEESO]
Reactive systems and nondeterminism: The real-world systems treated in this book usually contain at least one digital component which controls the oper- ation of other analog or digital components, and the whole system reacts to stimuli triggered by its environment As an example, consider a computer communication network in which components like routers, switches, hubs, and communication lines fulfill the common task of transferring data packets between the various computers connected to the network If the system of interest is the communication network only, the connected computers can be regarded as its environment which triggers the network by sending and receiv- ing data packets The behavior of the systems studied in this book can be characterized as nondeterministic since the stimulation by the environment is usually unpredictable In case of a communication system like the Internet,
a specific user will start to access information on the WWW via a browser
is usually riot known in advance Another source of nondeterminism is the potential failure of one or several system components, which in most cases leads to an altered behavior of the complete system
Modeling vs Measurement: In contrast to the empirical methods of measure- ment, i.e., the collection of output data during the observation of an executing system, the deductive methods of model-based performance evaluation have the advantage to be applicable in situations when the system of interest is not yet existing Deductive methods can thus be applied during the early design phases of the system developnient process in order to ensure that the final product meets its performance and reliability requirements Although the material presented in this book is restricted to modeling approaches, it should
be noticed that measurement as a supplementary technique can be employed
to validate that the conclusions obtained by model-based performance evalu- ation can be translated into useful statements about the real-world system Another possible scenario for the application of modeling is the situation
in which measurements on an existing system would either be too dangerous
or too expensive New policies, decision rules, or information flows can be explored without disrupting the ongoing operation of the real system More- over, new hardware architectures, scheduling algorithms, routing protocols,
or reconfiguration strategies can be tested without committing resources for their acquisition/implementation Also, the behavior of an existing system
Trang 24be evaluated before starting with the formalization process Here, the appli-
in a concrete modeling exercise
As an illustrative example, consider the power outage problem of computer systems For a given hardware configuration, there is no ideal way to repre- sent it without taking into consideration the software applications which run
on the hardware and which of course have to be reflected in the model In a real-time context, such as flight control, even the shortest power failure might have catastrophic implications for the system being controlled Therefore, an appropriate reliability model of the flight control computer system has to be very sensitive to such a (hopefully) rare event of short duration In contrast, the total number of jobs processed or the work accomplished by the com- puter hardware during the duration of a flight is probably a less important performance measure for such a safety-critical system If the same hardware configuration is used in a transaction processing system, however, short out- ages are less significant for the proper system operation but the throughput
is of predominant importance As a consequence thereof, it is not useful to represent effects of short interruptions in the model, since they are of less importance in this application context
Another important aspect to consider at the beginning of a model-based evaluation is how a reactive real-world system - as the core object of the study - is triggered by its environment The stimulation of the system by its environment has to be captured in such a way during formalization so
it reflects the conditions given in the real world as accurately as possible Otherwise, the measures obtained during the evaluation process cannot be meaningfully retrarisformed into statements about the specific scenario in the application domain In t,he context of stochastic modeling, the expression of the environment’s influence on the system in the model is usually referred to
of the arriving workload, e.g., the parts which enter a production line or the arriving data packets in a communication system, as a stochastic arrival
scenarios can be defined (see Section 6.8)
Trang 25METHODOLOGICAL BACKGROUND 7
The following four categories of system properties which are within the scope of the methods presented in this book can be identified:
Performance Properties: They are the oldest targets of performance evalua-
tion and have been calculated already for non-compiiting systems like tele- phone switching centers [Erlal7] or patient flows in hospitals [Jack541 using closed-form descriptions from applied probability theory Typical properties
to be evaluated are the mean throughput of served customers, the mean wait- ing, or response time and the utilization of the various system resources The IEEE standard glossary of software engineering terminology [IEEESO] con- tains the following definition:
Definition 1.1 Performance: The degree to which a system or component accomplishes its designated functions within given constraints, such as speed, accuracy, or memory usage
Reliability and Availability: Requirements of these types can be evaluated
quantitatively if the system description contains information about the fail- ure and repair behavior of the system components In some cases it is also necessary to specify the conditions under which a new user cannot get access
to the service offered by the operational system The information about the failure behavior of system components is usually based on heuristics which are reflected in the parameters of probability distributions In [IEEESO], software reliability is defined as:
Definition 1.2 Reliability: The probability that the software will not cause the failure of the system for a specified time under specified conditions System reliability is a measure for the continuity of correct service, whereas availatdity measures for a system refer to its readiness for correct service, as stated by the following definition from [IEEESO]:
Definition 1.3 Availability: The ability of a system to perform its required function at a stated instant or over a stated period of time It is usually expressed as the availability ratio, i.e., the proportion of time that the service
is actually available for use by the Customers within the agreed service hours Note that reliability and availability are related yet distinct system properties:
a system which - during a mission time of 100 days -~ fails on average every two minutes but becomes operational again after a few milliseconds is not very reliable but nevertheless highly available
Dependability and Performability: These terms and the definitions for them
originated from the area of dependable and fault tolerant computing The
following definition for dependability is taken from [ALRL04]:
Definition 1.4 Dependability: The dependability of a computer system
is the ability to deliver a service that can justifiably be trusted The service
Trang 26delivered by a system is its behavior as it is perceived by its user(s); a user
is another system (physical, human) that interacts with the former at the service interface
This is a rather general definition which comprises the five attributes availabil- ity, reliability, maintainability - the systems ability to undergo modifications
or repairs, integrity - the absence of improper system alterations and safety
as a measure for the continuous delivery of service free from occurrences of catastrophic failures The term performability was coined by J.F MEYER [Meye781 as a measure to assess a system’s ability to perform when perfor- mance degrades as a consequence of faults:
Definition 1.5 Performability: The probability that the system reaches
an accomplishment level y over a utilization interval (0, t ) That is, the prob- ability that the system does a certain amount of useful work over a mission time t
Subsequently, many other measures are included under performance as we
shall see in Section 2.2 Informally, the performability refers to performance
in the presence of failures/repair/recovery of components and the system Performability is of special interest for gracefully degrading systems [Beau77]
In Section 2.2, a framework based on Markov reward models (MRMs) is pre-
sented which provides recipes for a selection of the right model type and the definition of an appropriate performance measure
1.2.2 The Modeling Process
The first step of a model-based performance evaluation consists of the formal- ization process, during which the modeler generates a formal description of
the real-world system Figure 1.2 illustrates the basic idea: Starting from an
informal system description, e.g in natural language, which includes struc- tural and functional information as well as the desired performance and reli- ability requirements, the modeler creates a formal model of the real-world system using a specific conceptualization A conceptualization is an abstract, simplzfied view of the reference reality which is represented for some purpose
Two kinds of conceptualizations for the purpose of performance evaluation are presented in detail in this book: If the system is to be represented as a queue- ing network, the modeler applies a ?outed job flow” modeling paradigm in
which the real-world system is conceptualized as a set of service stations which are connected by edges through which independent entities “ f l o ~ ” through the network and sojourn in the queues and servers of the service stations (see Chapter 7) In an alternative Markov chain conceptualization a “state- transition” modeling paradigm is applied in which the possible trajectories through the system’s global state space are represented as a graph whose directed arcs represent the transitions between subsequent system states (see
Chapter 2) The main difference between the two conceptualizations is that
Trang 27Fig 1.2 Formalization of a real-world system
During the formalization process the following abstractions with respect to the real-world system are applied:
0 In both conceptualizations the behavior of the real-world system is regarded to evolve in a discrete-event fashion, even if the real-world system contains components which exhibit continuous behavior, such as the movements of a conveyor belt of a production line
0 The application of the queueing network formalism abstracts away from
system If the representation of these synchronization mechanisms is crucial in order to obtain useful results from the evaluation, the niodeler can resort to variants of stochastic Petri nets as an alternative descrip- tion t,echnique (see Section 2.3 and Section 2.3.6) in which almost arbi- trary synchronization pattcrns can be captured
association of system activity durations with random variables and the inclusion of branching probabilities to represent alternative system evo- lutions Both abstractions resolve the nondeterminisrn inherent in the
Trang 2810 lNTRODUCTlON
real-world system and turn the formal queueing network or Markov chain prototype into an “executable” specification [WingOl] For these,
at any moment during their operation each possible future evolution has
a well-defined probability to o c ( w Depending on which kind of ran- dom variables are used to represent the durations of the system activities either a discrete-time interpretation using a DTMC or a continuous-time interpretation of the system behavior based on a CTMC is achieved It should be noted that for systems with asynchronously evolving com- ponents the continuous-time interpretation is more appropriate since changes of the global system state may occur at any moment in contin- uous time Systems with components that evolve in a lock-step fashion triggered by a global clock are usually interpreted in discrete-time
1.2.3 Evaluation
The second step in the model-based system evaluation is the deduction of performance measures by the application of appropriate solution methods Depending on the conceptualization chosen during the formalization process the following solution methods are available:
Analytical Solutions: The core principle of the analytic: solution methods is
to represent the formal system description either as a single equation from which the interesting measures can be obtained as closed-form solutions, or
as a set of system equations from which exact or approximate measures can
be calculated by appropriate algorithms from numerical mathematics
1 Closed-form solutions are available if the system can be described as
a simple queucing system (see Chapter 6) or for simple product-form queueing networks (PFQN) [Chhla83] (see Section 7.3) or for structured small CTMCs For these kind of formalizations equations can be derived from which the mean number of jobs in the service stations can be calculated as a closed-form solution, i.e., the solutions can be expressed analytically in terms of a bounded number of well-known operations Also from certain types of Markov chains with regular structure (see Section 3 1), closed-form representations like the well-known Erlang-
B and Erlang-C formulae [Erlal7] can be derived The measures can either be computed by ad-hoc programming or with the help of computer algebra packages such as Mathernatica [Mat05] A big advantage of‘the closed-form solutions is their moderate computational complexity which enables a fast calculation of performance measures even for larger system descriptions
from a formal system description do not possess a closed-form solution, e.g., in the case of complex systems of integro-differential equations
In these cases, approximate solutions can be obtained by the appli-
Trang 29METHODOLOGICAL BACKGROUND 11
cation of algorithms from numerical mathematics, many of which are implemented in computer algebra packages [Mat051 or are integrated in performance analysis tools such as SHARPE [HSZTOO], SPNP [HTTOO],
or TimeNET [ZFGHOO] (see Chapter 12) The formal system descrip- tions can be either given as a queueing network, stochastic Petri net or another high-level modeling formalism, from which a state-space repre- sentation is generated manually or by the application of state-space gen- eration algorithms Depending on the stochastic information present in the high-level description, various types of system state equations which mimic the dynaniics of the modeled system can be derived and solved
by appropriate algorithms The numerical solution of Markov models
is discussed in Chapters 3 ~ 5, numerical solution methods for queueing networks can be found in Chapters 7 - 10 In comparison to closed-form solution approaches, numerical solution met hods usually have a higher computational complexity
is feasible, because either a theory for the derivation of proper system equa- tions is not known, or the computational complexity of an applicable numeri- cal solution algorithm is too high In this situation, solutions can be obtained
by the application of discrete-event simulation (DES), which is described in detail in Chapter 11 Instead of solving system equations which have been derived from the formal model, the DES algorithm “executes” the model and collects the information about the observed behavior for the subsequent derivation of performance measures In order to increase the quality of the results, the simulation outputs collected during multiple “executions” of the model are collected and from which the interesting measures are calculated by statistical methods All the formalizations presented in this book, i.e., queue- ing networks, stochastic Petri nets, or Markov chains can serve as input for
a DES, which is the most flexible and generally applicable solution method Since the underlying state space does not have to be generated, simulation
is not affected by the state-space explosion problem Thus, simulation can also be employed for the analysis of complex models for which the numerical approaches would fail because of an exuberant number of system states
eling formalisms and solution methods are combined in oder to exploit their coniplementing strengths Examples of hybrid solution methods are mixed simulation and analytical/numerical approaches, or the combination of fault trees, reliability block diagrams, or reliability graphs, and Markov models [STP96] Also product-form queueing networks and stochastic Petri nets
or non-product-form networks and their solution methods can be combined More generally, this approach can be characterized as intermingling of state- space-based and non-state-space-based methods [STP96] A combination of analytic and simulative solutions of connected sub-models may be employed
Trang 30to combine the benefits of both solution methods [Sarg94, ShSa831 More cri- teria for a choice between simulation and analytical/numerical solutions are discussed in Chapter 11
tems, generalized stochastic Petri nets (GSPNs), and stochastic reward nets (SRNs), as the most prominent representatives, have been suggested in the literature to automate the model generation [HaTr93] GSPNs/SRNs that are covered in more detail in Section 2.3, can be characterized as tolerating large- ness of the underlying computational models and providing effective means for generating large state spaces
Largeness Avoidance: Another way to deal with large models is to avoid the creation of such models from the beginning The major largeness-avoidance technique we discuss in this book is that of product-form queueing networks The main idea is, the structure of the underlying CTMC allows for an efficient solution that obviates the need for generation, storage, and solution of the large state space The second method of avoiding largeness is to separate the originally single large problem into several smaller problems and to combine sub-model results into an overall solution Both approximate and exact tech- niques are known for dealing with such multilevel models The flow of informa- tion needed among sub-models may be acyclic, in which case a hierarchical model [STP96] results If the flow of needed information is non-acyclic, a fixed-point iteration may be necessary [CiTr93] Other well-known techniques applicable for limiting model sizes are state truncation [BVDT88, GCS+86]
1.2.4 Summary
Figure 1.3 summarizes the different phases and activities of the model-based performance evaluation process Two main scenarios are considered: In the first one, model-based performance evaluation is applied during the early phas-
es of the system development process to predict the performance or reliability properties of the final product If the predicted properties do not fulfill the given requirements, the proposed design has to be changed in order to avoid the expected performance problems In the second scenario, the final prod- uct is already available and model-based performance evaluation is applied to derive optimal system configuration parameters, to solve capacity planning problems, or to check whether the existing system would still operate satis- factorily after a modification of its environment In both scenarios the first activity in the evaluation process is to collect information about the structure and functional behavior of an existing or planned system The participation
of a domain expert in this initial step is very helpful and rather indispensable for complex applications Usually, the collected information is stated infor- mally and stored in a document using either a textual or a combined textu-
Trang 31METHODOLOGICAL BACKGROUND 13
al/graphical form According to the goal of the evaluation study, the informal description also contains the performance, reliability, availability, dependabil- ity, or performability requirements that the system has to fulfill Based on an informal system description, the formalization process is carried out by the modeler who needs both modeling and application-specific knowledge
Fig 1.3
ment process and configurat,ion management
Application of model-based performance evaluation in the system develop-
Trang 3214 lNTRODUCTlON
If he selects a Markov chain conceptualization; a formal representation on the state-space level is created If he chooses instead a queueing network or stochastic Petri net concept,ualization, a formal high-level model is derived The formal model represents the system as well as the interaction with its environment on a conceptual level and, as a result, the model abstracts from all details which are considered to be irrelevant for the evaluation In the scenario of an existing real-world system, a conceptual validation of the cor- rectness of the high-level model can be accomplished in an iterative process
of step-wise refinement [Wirt71, Morr871, which is not represented in Fig 1.3
Conceptual validation can be further refined [NaFi67] into “face validation,” where the involved domain experts try to reach consensus on the appropriate- ness of the model on the basis of dialogs, and into the “validation of the model assumptions,” where implicit or explicit assumptions are cross-checked Some
of the crucial properties of the model are checked by answering the following questions [HMTSl]:
0 Is the model logically correct, complete, or overly detailed?
0 Are the distributional assumptions justified? How sensitive are the results to simplifications in the distributional assumptions?
0 Are other stocha.stic properties, such as independence assumptions, valid?
0 Is the model represented on the appropriate level? Are those features included that are most significant for the application context?
The deduction of performance measures can be carried out either by the application of closed-form solution methods/simulation based on a high-level description or by numerical analysis/simulation of a Markov chain The link between high-level model and semantic model represents the state-space gen- eration activity which is in most cases performed automatically inside an appropriate performance analysis tool The use of tools for an automated generation of state-space representations has the advantage of generating a semantic model that can be regarded as a lower-level implementation reflect- ing all system properties as described in the high-level model
data collected through measurements, if an existing system is available The
validation results are used for modification of the high-level model In the sys- tem development scenario the calculated performance measures are used to verify whether the final product meets its performance related requirements
If the answer from the verification step is “NO,” the formal system description has to be redesigned and the performance evaluation process starts anew If the answer of the verification step is “YES,” the system development process can move on to the subsequent detailed design and implementation phases
In the system configuration and capacity planning scenarios there is a strong relation between measurements and modeling Measurement data are
to be used for model validation Furthermore, model parameterization relies After performance measures have been derived, they can be validated against
Trang 33BASICS O f PROBABILITY AND STATISTICS 15
heavily on input from measurement results Model parameters are frequent-
ly derived from measurements of earlier studies on similar systems also iri the system development scenario Conversely, measurement studies can often
be better planned and executed if they are complemented and guided by a model-based evaluation
1.3 BASICS OF PROBABILITY AND STATISTICS
We begin by giving a brief overview of the more important definitions and results of probability theory The reader can find additional details in books such as [Allego, Fe1168, Koba78, TrivOl] We assume that the reader is familiar with the basic properties and notations of probability theory
1.3.1 Random Variables
A random variable is a function that reflects the result of a random experi-
ment For example, the result of the experiment “toss a single die” can be described by a random variable that can assume the values one through six The number of requests that arrive at an airline reservation system in one hour
or the number of jobs that arrive at a computer system are also examples of
a random variable So is the time interval between the arrivals of two consec- utive jobs at a computer system, or the throughput in such a system The latter two examples can assume continuous values, whereas the first two only assume discrete values Therefore, we have to distinguish between continuous and discrete random variables
discrete values is called a discrete random variable, where the discrete values
are often non-negative integers The random variable is described by the pos- sible values that it can assume and by the probabilities for each of these values The set of these probabilities is called the probability mass function (pmf) of
this random variable Thus, if the possible values of a random variable X are
the non-negative integers, then the pmf is given by the probabilities:
p k = P ( x = k ) , for k = 0 , 1 , 2 I (1.1) the probability that the random variable X assumes the value k
The following is required:
Trang 3416 lNTRODUCTlON
The following are other examples of discrete random variables:
0 Bernoulli random variable: Consider a random experiment that has two possible outcomes, such as tossing a coin ( k = 0 , l ) The pmf of the random variable X is given by
P ( X = 0) = 1 - p and P ( X = 1) = p , with 0 < p < 1 (1.2)
0 Binomial random variable: The experiment with two possible outcomes
is carried out n times where successive trials are independent The random variable X is now the number of times the outcome 1 occurs The pmf of X is given by
p k ( l - p)"-" k = 0,1, ,n (1.3)
0 Geometric random variable: The experiment with two possible outcomes
is carried out several times, where the random variable X now represents the number of trials it t,akes for the outcome 1 to occur (the current trial included) The pmf of X is given by
The Poisson and geometric random variables are very important to our topic;
we will encounter them very often Several important parameters can be derived from a pmf of a discrete random variable:
0 Mean value or expected value:
Trang 35BASICS OF PROBABILITY AND STATISTICS 17
that is, the nth moment is the expected value of the nth power of X The first moment of X is simply the mean of X
where cx is called the standard deviation
0 The coefficient of variation is the normalized standard deviation:
Table 1.1 Properties of several discrete random variables
Trang 3618 INTRODUCTION
all values in the interval [a, b], where -cx 5 (L < b 5 foo, is called a contin-
CDF or cumulative distribution function):
Continuous Random Variables
Fx(X) = P ( X I x) 1 (1.13) which specifies the probability that the random variable X takes values less than or equal to x, for every x
From Eq (1.13) we get for x < y:
The probability density function (pdf) f x ( x ) can be used instead of the dis- tribution function, provided the latter is differentiable:
(1.14) Some properties of the pdf are
0 Mean value or expected value:
(1.15)
Trang 37BASICS OF PROBABILITY AND STATISTICS 19
0 Coefficient of variation:
C X = r O X (1.20)
X
A very well-known and important continuous distribution function is the
The standard normal distribution is defined by setting fT; = 0 and ux = 1:
A plot of the preceding pdf is shown in Fig 1.4
For an arbitrary normal distribution we have
(1.22)
respectively
Other important continuous random variables are described as follows
Trang 3820 INTRODUCTION
X
fig 1.4 pdf of the standard normal random variable
(a) Exponential Distribution
The exponential distribution is the most important and also the easiest to use distribution in queueing theory Interarrival times and service times can often be represented exactly or approximately using the exponential distribu- tion The CDF of an exponentially distributed random variable X is given
by Eq (1.23):
otherwise
(1.23)
- if X represents interarrival times,
if X represents service times
with =
- ,
Here X or p denote the parameter of the random variable In addition, for
an exponentially distributed random variable with parameter X the following relations hold:
pdf: fx (x) = A ePX" ,
- 1 mean: X = -
A '
1 X*
variance: var(X) = - , coefficient of variation: c x = 1 Thus, the exponential distribution is completely determined hy its mean value
Trang 39BASICS OF PROBABILITY AND STATISTICS 21
The importance of the exponential distribution is based on the fact that it
is the only continuous distribution that possesses the memoryless property:
P ( X 5 u + t I X > u) = 1 - exp -= = P ( X 5 t ) (1.24)
As an example for an application of Eq (1.24), consider a bus stop with the following schedule: Buses arrive with exponentially distributed interarrival times and identical mean x Now if you have already been waiting in vain for u units of time for the bus to come, the probability of a bus arrival within the next t units of time is the same as if you had just shown up at the bus stop, that is, you can forget about the past or about the time already spent waiting
Another important property of the exponential distribution is its relation
to the discrete Poisson random variable If the interarrival times are expo- nentially distributed and successive interarrival times are independent with identical mean x, then the random variable that rcpresents the number of buses that arrive in a fixed interval of time [ O , t ) has a Poisson distribution with parameter cy = t/x
Two additional properties of the exponential distribution can be derived from the Poisson property:
( 3
1 If we merge n Poisson processes with distributions for the interarrival times 1 - e-’it, 1 5 i 5 n, into one single process, then the result is
a Poisson process for which the interarrival times have the distribution
1 - e-’t with X = cy=l X i (see Fig 1.5)
fig 1.5 Merging of Poisson processes
2 If a Poisson process with interarrival time distribution 1 - ePxt is split into n processes so that the probability that the arriving job is assigned
to the ith process is q2, 1 5 i 5 n, then the ith subprocess has an interarrival time distribution of 1 - e-qiXt, i.e., R Poisson processes have been created, as shown in Fig 1.6
The exponential distribution has many useful properties with analytic trac- tability, but is not always a good approximation to the observed distribution Experiments have shown deviations For example, the coefficient of variation
of the service time of a processor is often greater than one, and for a peripheral device it is usually less than one This observed behavior leads directly to the need to consider the following other distributions:
Trang 40Fig 1.6 Splitting of a Poisson process
(b) Hyperexponential Distribution, HI,
This distribution can be used to approximate empirical distributions with a coefficient of variation larger than one Here k is tho number of phases
Fig 1.7 A random variable with H k distribution
Figure 1.7 shows a model with hyperexponentially distributed time The model is obtained by arranging k phases with exponcmtially distributed times and rates p l , pz, , pk in parallel The probability that the time span is
given by the j t h phase is qrr, where c:=, q3 = 1 However, only one phase can be occupied at any time The resulting CDF is given by
In addition, the following relations hold:
(1.25)